# cylindrical or spherical coordinates problem

• Nov 20th 2008, 09:59 AM
iwonder
cylindrical or spherical coordinates problem
How to evaluate:
∫∫∫ xyz/(x^2+y^2)^1/2 dv
E

where E is the solid "quarter-cylinder" in the first octant bounded by a cylinder of radius a>o with axis long the z axis, the planes y=0,x=0,z=0 and z=b, wiht b>0.

Thankyou very much!
• Nov 20th 2008, 11:23 AM
JD-Styles
The best way to tackle this one is to change to cylindrical coordinates. So we'd get:

$\displaystyle \int_0^b \int_0^{\pi /2} \int_0^a \frac{(\rho cos\phi )(\rho sin\phi ) z}{\rho } \rho d\rho d\phi dz$

$\displaystyle \int_0^b zdz \int_0^{\pi /2} cos\phi sin\phi d\phi \int_0^a \rho ^2 d\rho$

$\displaystyle (\frac{b^2}{2})(\frac{1}{2})(\frac{a^3}{3})$

$\displaystyle \frac{a^3 b^2}{12}$
• Nov 20th 2008, 09:58 PM
iwonder
thankyou!
i got it so far, but how do u get p^2 for the third limit of integration? is that only be p???
• Nov 23rd 2008, 09:22 AM
JD-Styles
There's one p from x, on from y, one from the change to spherical coordinates (that comes with d phi), so that's p^3 in the numerator. Then in the denominator you have one p from (x^2+y^2)^1/2. So in total you have p^(3-1)=p^2